
Now by Levy's Theorem (cf. Theorem 1.7.6 [1]}, an inspection of the proof reveals that in the univariate case convergence almost everywhere is a sufficient condition), we conclude that \(\hat{F}_{I,n}\rightarrow F_I\)  and \(\hat{F}_{G,n}\rightarrow F_G\)  point-wise for all realisations in \(\Omega \) . Since the distribution functions are continuous, these convergences are also uniform. Recall that the above argument holds for a subsequence and all realisations in \(\Omega \) . But we can repeat the same argument starting from a subsequence and showing in that way that every subsequence contains a sub-subsequence for which the corresponding distribution functions converge almost surely. This implies convergence in probability by Lemma 1.9.2 in [2]} and the proof is complete.
